Gravitation Class 11 notes Physics Chapter 8

Introduction

Have you ever wondered, why does anything thrown up fall down? Why does not the moon fall to the earth? Why does not the earth fly off into space rather than remaining in orbit around the sun? Why do all the planets revolve around the sun?

Kepler’s laws

In astronomy,Kepler’s laws  of planetary motion, proposed by Johannes Kepler during the period of 1609 to 1619, describe the orbits of planets around the Sun. The motion of planets has been a subject of much interest for astronomers from very early times.

(i) Kepler’s First Law (The Law of Orbits)

It states that “All planets move around the sun in elliptical orbits with sun at one of the focus not at centre of orbit.”

(ii) Kepler’s Second Law (The Law of Areas)

Kepler’s second law states that “The line joining the sun and planet sweeps out equal areas in equal time or the rate of sweeping area by the position vector of the planet with respect to sun remains constant.”

Let P be the instantaneous position of planet relative to sun. The position vector of P relative to sun is r. After time Δt, the planet is at Q, having position vector r+Δr such let Δr = PQ

The area swept out by radius SP in time Δt.

ΔA = Area of triangle PSP’

ΔA=12r×ΔrΔA=12r×Δr

Area swept by radius vector per second or Areal velocity,

dAdt=limΔt→0ΔAΔtdAdt=limΔt→0ΔAΔt

dAdt=limΔt→012r×ΔrΔtdAdt=limΔt→012r×ΔrΔt

dAdt=12r×drdtdAdt=12r×drdt

dAdt=12r×vdAdt=12r×v

r×v=Jm=constantr×v=Jm=constant

Areal velocity,

dAdt=12⋅Jm=constantdAdt=12⋅Jm=constant

It follows that the are velocity of the radius vector of planet relative to sun remains constant. This is Kepler’s Second Law.

(iii) Kepler’s Third Law (The Law of Periods)

It states that “The time period of revolution of a planet in its orbit around the sun is directly proportional to the cube of semi-major axis of the elliptical path around the sun.”

Let T be period of revolution of the planet in elliptical orbit, area of ellipse = πab a and b being semi-major and semi-minor axes of the ellipse.

Areal velocity,

dAdt=J2mdAdt=J2m

∴ Period of revolution

T=Area−of−ellipseAreal−velocityT=Area-of-ellipseAreal-velocity

T=πabJ2mT=πabJ2m

T2=4π2m2a2b2J2T2=4π2m2a2b2J2

If l is the semi-latus rectum of ellipse, then

l=b2al=b2a

T2=4π2m2a2laJ2T2=4π2m2a2laJ2

i.e.,

T2∝a3T2∝a3

Since all other quantities are constant. This is Kepler’s third law.

Universal law of gravitation

According to this law, “Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them”.

Mathematically,

→F∝m1m2F→∝m1m2 …..(1)

→F∝1r2F→∝1r2 …..(2)

From eq. (1) & (2), we get

→F∝m1m2r2F→∝m1m2r2

→F=Gm1m2r2F→=Gm1m2r2

where G is the universal gravitational constant.

Acceleration Due to Gravity of the Earth

(i) Above the surface of the Earth

Let us consider a point mass m at a height h above the surface of the earth.

Assume the radius of earth be R_e and the mass of earth be M_e.

Now, as we know that the mass of earth is concentrated at its centre. The magnitude of force on the point mass m will be

Fh=GMem(Re+h)2Fh=GMem(Re+h)2 …..(i)

The acceleration experienced by that point mass will be

gh=Fm=GMem(Re+h)2mgh=Fm=GMem(Re+h)2m

gh=GMe(Re+h)2gh=GMe(Re+h)2

From the above equation it is clear that as we move above the surface of earth the value of g reduces.

(ii) Below the surface of the Earth

Let us consider a point mass m at A which is present at depth d below the surface of the earth. Consider the radius of earth as R_e and a mass of M_e.

As per our discussion, we know that the gravitational force on point mass m will be due to the smaller sphere of radius (R_e–d) whose mass M_s (say) is concentrated at the centre.

Now,

Ms=43π(Re−d)3ρMs=43π(Re-d)3ρ

Me=43πR3eρMe=43πRe3ρ

MsMe=[Re−dRe]3MsMe=[Re-dRe]3

By universal law of gravitation

Fd=GMem(Re−d)R3eFd=GMem(Re-d)Re3

As we know from Newton’s second law that

gd=Fmgd=Fm

gd=GMeRe−dR3gd=GMeRe-dR3

gd=gRe−dRegd=gRe-dRe …..[∴g=GMeR2e∴g=GMeRe2]

gd=g[1−dRe]gd=g[1-dRe]

Thus, as we go down below the earth’s surface, the acceleration due to gravity decreases.

Gravitational potential energy

The potential energy of a system corresponding to a conservative force has been defined as

Uf−Ui=−∫F.drUf-Ui=-∫F.dr

Let a particle of mass m₁ and another particle of mass m₂ is taken, then The force on the particle of mass m₂ is

F=Gm1m2r2F=Gm1m2r2

The work done by the gravitational force in the displacement from r to r + dr is

dW=−Gm1m2r2drdW=-Gm1m2r2dr

W=−∫∞rGm1m2r2drW=-∫r∞Gm1m2r2dr

W=−Gm1m2[−1r]∞rW=-Gm1m2[-1r]r∞

W=−Gm1m2(−1∞+1r)W=-Gm1m2(-1∞+1r)

U=−Gm1m2rU=-Gm1m2r

Escape speed

It is the minimum velocity with which a body should be projected from the surface of a planet so as to reach infinity. Let a body of mass m be projected from the earth’s surface with speed u. The total  remains conserved i.e.,

Total mechanical energy on earth = Total mechanical energy at infinity

Ki+Ui=Kf+UfKi+Ui=Kf+Uf

12mv2e−GMmR=0+012mve2-GMmR=0+0

ve=√2GMRve=2GMR

ve=√2gRve=2gR …..[∴GM=gR2∴GM=gR2]

On the earth’s surface, v_e = 11.2 km/s

Earth satellites

Satellite is a body which revolves continuously in an orbit around a comparatively much larger body. Earth satellites are those objects which revolve around the earth. The motion of satellites is very similar to the motion of planets around the sun and hence Kepler’s laws of planetary motion are most likely applicable to them. Moon is the only natural satellite of the earth with nearly a circular orbit with the time period of approximately 27.3 days which is also roughly equal to the rotational period of the moon about its own axis.

Geostationary and polar satellites

Suppose a satellite on equatorial plane of the earth is revolving with a time period of 24 hours then it would appear stationary from a point on earth. Hence, the satellite which appears at a fixed position and at a definite height to an observer on earth, is called a geostationary satellite.

Weightlessness

Weight of an object is the force with which the earth attracts it. We are conscious of our own weight when we stand on a surface, since the surface exerts a force opposite to our weight to keep us at rest.

“When an object is in free fall, it is weightless and this phenomenon is usually called the weightlessness.” In a manned satellite, people inside experience no gravity. This is a example of weightlessness.

Summary

• Gravitational force: It is a force of attraction between the two bodies by the virtue of their masses.
• Acceleration due to gravity: The acceleration produced in the motion of a body freely falling towards earth under the force of gravity is known as acceleration due to gravity.
• Gravitational potential energy: The amount of work done in displacing the particle from infinity to a point under consideration.
• Gravitational potential: The gravitational potential due to the gravitational force of the earth is defined as the potential energy of a particle of unit mass at that point.
• Escape speed: The minimum speed with which the body has to be projected vertically upwards from the surface of the earth is called escape speed.
• Orbital speed: The minimum speed required to put the satellite into the given orbit around earth is called orbital speed.
• Satellite: It is a body which revolves continuously in an orbit around a comparatively much larger body.
• Polar satellite: It is the satellite which revolves in polar orbit around the earth.
• Geostationary satellite: It is the satellite which appears at a fixed position and at a definite height to an observer on earth.
• Kepler’s Ist law: All planets move in elliptical orbits, with the sun at one of foci of the ellipse.
• Kepler’s IInd law: The line that joins any planet to the sun sweeps out equal areas in equal intervals of time.
• Kepler’s IIIrd law: The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet.
• Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Gravitational force is a conservative force.
• The value of acceleration due ot gravity is maximum at the surface of the earth while zero at the centre of earth.
• Henry Cavendish was the first person who found the value of G experimentally.
• Gravitational force on a particle inside a spherical shell is zero.
• Gravitational shielding is not possible.
• An astronaut experiences weightlessness in a space satellite. It is because both the astronaut and the satellite are in “free fall” towards the earth.
• The value of g increases from equator to poles.
• The escape speed from a point on the surface of the earth may depend on its location on the earth e.g., escape speed is more on poles and less on equator.
• The orbital speed of satellite is independent of mass of the satellites.
• Kepler’s laws hold equally well for satellites.